Quadratic Equations — Quadratic Formula, Discriminant, and Examples

A quadratic equation is an equation in the form $ax^2 + bx + c = 0$, where the highest power of the unknown variable is $2$. For example, $x^2 - 4x + 1 = 0$ is a quadratic equation, whereas $2x + 3 = 0$ is a linear equation. Keep these two key formulas in mind from the start:

$ax^2 + bx + c = 0 \quad (a \ne 0)$

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here $a$, $b$, and $c$ are the coefficients. In graphing terms, solving the equation is equivalent to finding where the parabola $y = ax^2 + bx + c$ intersects the $x$-axis, which is why the discriminant is so useful for predicting the number of solutions.

Why the discriminant comes first

The discriminant is defined as:

$D = b^2 - 4ac$

Since $D = b^2 - 4ac$ sits inside the square root of the quadratic formula, its sign directly determines how many solutions exist within the real numbers:

• If $D > 0$, there are two distinct real roots.
• If $D = 0$, there is one repeated root (double root).
• If $D < 0$, there are no real roots.

The important condition is the "real number range." If we expand our scope to complex numbers, solutions can still exist even if $D < 0$. Checking the discriminant first tells you exactly how many solutions to expect before you start the main calculation.

The calculation procedure

For a test or homework, follow three steps: organize the equation into standard form, use the discriminant to determine the number of solutions, then calculate with the quadratic formula if necessary.

If an equation factors easily, factoring is the fastest method. But exams often include equations that don't break down neatly into integers, and in those cases the quadratic formula is the most reliable:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Ensure the equation is organized into $ax^2 + bx + c = 0$ before using this formula. Plugging in numbers without organizing first makes it easy to mistake the signs of $b$ or $c$.

Worked example: from start to finish

Solve

$x^2 - 4x + 1 = 0$

Since it is already in standard form:

$a = 1,\quad b = -4,\quad c = 1$

1. Check the discriminant first.

$D = b^2 - 4ac = (-4)^2 - 4(1)(1) = 16 - 4 = 12$

Since $D = 12 > 0$, there are two distinct real solutions.

2. Substitute into the quadratic formula.

$x = \frac{-(-4) \pm \sqrt{12}}{2 \cdot 1}$

Simplifying:

$x = \frac{4 \pm \sqrt{12}}{2}$

Since $\sqrt{12} = 2\sqrt{3}$:

$x = \frac{4 \pm 2\sqrt{3}}{2}$

Therefore:

$x = 2 \pm \sqrt{3}$

Two takeaways: you can solve the problem even when factoring isn't obvious, and checking the discriminant first tells you how many solutions to expect before the main calculation.

Practice on your own

Following the same steps, try:

$2x^2 + 3x - 1 = 0$

First identify $a$, $b$, and $c$, calculate the discriminant, and finish with the quadratic formula. Solving one more similar problem makes the process much clearer.

Calculation traps to avoid

Plugging numbers in without converting to standard form

You cannot plug $x^2 + 1 = 4x$ directly into the formula. Move it to

$x^2 - 4x + 1 = 0$

first, so that you correctly identify $b = -4$.

Mistakes with the sign of $-b$

If $b = -4$, then $-b = 4$. A single sign error here changes your entire final answer.

Calculating the discriminant but forgetting what it means

Remember the connection: $D > 0$ means two solutions, $D = 0$ means one, and $D < 0$ means no real roots. The discriminant isn't just a tool for calculating a number; it reads the nature of the solutions.

Where quadratic equations are used

Quadratic equations are the foundation of middle and high school algebra and also appear constantly in parabolic graphs, area problems, and maximum/minimum value problems. Whenever a situation is modeled with a squared term, you will eventually need to solve a quadratic equation. When linked to function graphs, the solutions are the points where the parabola meets the $x$-axis, so the discriminant is a criterion for whether the graph intersects the $x$-axis twice, touches it once, or doesn't meet it at all.